# Schedule

Monday 28 Tuesday 29 Wednesday 30 Thursday 31 Registration Opening session D. Calaque B. Maury P. Omari J. Dolbeault Coffee break K. Polthier C. Geuzaine S. Jaffard A. Valette Lunch Game Theory Functional Analysis Submanifold Geometry Functional Analysis Algebraic Topology Algebra and applications Theory and applications of categories Algebra and applications Applied Analysis Numerical Schemes for PDEs Nonlinear elliptic PDEs Evolution PDEs and Applications Poster session& drink

The afternoon talks are 40 mins long. A coffee break is offered from 15:30 to 16:00. Sessions will normally consist of 4 talks. Some sessions (in particular, all sessions of Tuesday) will only have 3 talks and thus finish at 17:00.

On Monday and Tuesday, the talks will take place in Valenciennes and on Wednesday and Thursday in Mons. Transportation will be offered if you mention you need it when you register.

## Plenary talks

Please, click on the title to see the abstract.

• D. Calaque — Derived symplectic geometry and topological field theories.

I will start this talk with a summary of a very recent topic called derived symplectic geometry, reviewing in particular some results of Pantev-Toen-Vaquie-Vezzosi and myself on the construction of derived symplectic and Lagrangian structures on derived mapping stacks. I will then explain how this is related to the project of constructing various extended Topological Field Theories with values in an appropriate category of Lagrangian correspondences (inspired by the recent work of Cattaneo-Mnev-Reshetikhin).

• K. Polthier — Discrete geometry and global remeshing of surfaces and volumes.

Meshes arising from 3D scans or other 3D imaging techniques are often not well adjusted to the geometry or topology of the underlying shapes. The generation of good meshes is still an active research area in various disciplines. We discuss novel techniques to generate highly structured surface and volume meshes that are consistent with many geometric and topological features of the underlying shapes.

The presented mathematical techniques include discrete differential forms, discrete Hodge decomposition, and integrability issues which are fundamental also for a wide range of other local and global applications in geometry processing. Applications to various problems in scientific computing, computer aided design, architecture and computer graphics are shown.

• B. Maury — Congestion in crowd motion modeling.

Congestion is a key feature in the modeling of crowd motions. The term actually reflects two types of phenomena: social tendency to maintain a minimal distance with neighbors and, whenever there is actual contact, limited compressibility of the human body. In extreme situations, the second aspect is likely to be predominant: the crowd can be seen as an active granular medium.

This standpoint can be formalized in the microscopic context by identifying individuals to rigid disks, and defining a desired, or spontaneous, velocity field. The latter corresponds to what people would do if they were alone. This desired velocity field is then projected on the set of admissible fields (velocities that do not violate the non-overlapping constraint).

In the macroscopic setting, a similar approach can be carried out: the population is described by a density. It leads to an evolution problem of the transport type under maximal density constraint, that can be interpreted under some assumption as a gradient flow in the Wasserstein space of probability measures.

We shall describe a sound theoretical framework for both models, and address the numerical challenges they are raising (details in [1]).

[1] B. Maury, A. Roudneff-Chupin, F. Santambrogio, J. Venel, Handling Congestion in Crowd Motion Modeling, Networks and Heterogeneous Media, Volume 6, Number 3, September 2011, pp. 485–519, (arXiv:1101.4102 [math.NA]).
• C. Geuzaine — A Domain Decomposition Method for the Helmholtz Equation.

We present a domain decomposition method for the Hel­mholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions. A preconditioner based on the analytic inversion of the iteration operator is combined with this algorithm, which, for topologically 1-D decompositions, leads to convergence in a fixed number of iterations, independently of the frequency and the number of subdomains.

• P. Omari — Asymmetric Poincaré inequalities and existence of bounded variation solutions of the capillarity equation.

The aim of this talk is twofold: on the one hand we establish an asymmetric version of the classical Poincaré inequality in the space of bounded variation functions, on the other hand, based on this result, we prove, via elementary non-smooth critical point theory, the existence and the multiplicity of bounded variation solutions of a class of capillarity problems with asymmetric perturbations.

• S. Jaffard — Multifractal Analysis: A growing interface between mathematical analysis and signal processing.

Multifractal analysis was introduced by physicists with the purpose of using global regularity indices in order to study the intimate structure of fully developed turbulence. A later motivation was to explain the results observed as the signature of the presence of pointwise singularities of different strengths'' in the data. The heuristic arguments relating global and pointwise regularity which were introduced for that purpose proved extremely powerful inside mathematics, supplying unifying concepts with a surprisingly wide range of applications: Large classes of functions arising in harmonic analysis, number theory or stochastic processes have been revisited using this approach, and were shown to display fractal sets of singularities of varying Hausdorff dimensions. We will give some ideas of the functional and harmonic analysis tools that are involved in this theory, and present a selected choice of applications where multifractal analysis proved particularly relevant, including: Model classification in the context of fully developed turbulence, the diagnostic of heart-beat failure, modeling of internet flow, and stylometry tools in order to help art historians differentiate between the paintings of several masters. We will also mention some mathematical open problems motivated by recent applications [1].

[1] P. Abry, S. Jaffard, H. Wendt,Irregularities and scaling in signal and image processing: Multifractal analysis, Benoit Mandelbrot: A Life in Many Dimensions'' to appear, World Scientific, M. Frame Ed., 2013.
• J. Dolbeault — Interpolation inequalities: rigidity results, nonlinear flows and applications.

This lecture is motivated by interpolation inequalities of So­bo­lev type on smooth compact Riemannian manifolds without boundaries. The method is based on rigidity results for the solutions of an elliptic equation, which means that the PDE has no other solution than the constant one, at least when a parameter is in a certain range. This parameter can be used as an estimate for the best constant in the corresponding interpolation inequality.

Classical results are due to M.-F. Bidaut-Véron and L. Véron and extend an earlier work of B. Gidas and J. Spruck. The method will be illustrated by the simple case of the sphere [1]. In the case of a general manifold [5], the novelty is to exhibit a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of some technical choices of exponents done in earlier works. It also allows to extend results by J.R. Licois and L. Véron, D. Bakry and M. Ledoux, and J. Demange.

The main result relies on an integral criterion for rigidity, which is weaker than the classical pointwise conditions, and holds for general manifolds without uniform positivity condition on the curvature. Using the flow, it is then possible to discuss the optimality of the constant in the interpolation inequalities and get interesting consequences for spectral problems [2,4]. The method also allows to recover new properties of interpolation inequalities on the Euclidean space and produces flows of independent interest [3].

[1] J. Dolbeault, M. J. Esteban, M. Kowalczyk, and M. Loss, Sharp interpolation inequalities on the sphere: New methods and consequences, Chinese Annals of Mathematics, Series B, 34 (2013), pp. 99–112.
[2] J. Dolbeault, M. J. Esteban, and A. Laptev, Spectral estimates on the sphere, Analysis and PDE, (2013).
[3] J. Dolbeault, M. J. Esteban, A. Laptev, and M. Loss, One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows. In prep.
[4] , Spectral properties of Schrödinger operators on compact manifolds: rigidity, flows, interpolation and spectral estimates, C.R. Mathématique, (2013).
[5] J. Dolbeault, M. J. Esteban, and M. Loss, Nonlinear flows and rigidity results on compact manifolds, tech. rep., Ceremade, 2013.
• A. Valette — Distortion and spectral gap for finite graphs.

Let $$X$$ be a finite connected graph, viewed as a discrete metric space. The $$L^p$$-distortion of $$X$$ is a numerical invariant measuring the difficulty to embed $$X$$ isometrically into $$L^p$$. A fundamental result of Bourgain (1985) states that the $$L^p$$-distortion of $$X$$ is bounded above by (constant times) the logarithm of the number of vertices of $$X$$. We shall explain an application of this result to the algorithmic problem SPARSEST CUT from theoretical computer science. To conclude, we will explain a recent result (joint with P.-N. Jolissaint) providing a lower bound on the $$p$$-distortion of $$X$$ in terms of the $$p$$-spectral gap, i.e. the first non-zero eigenvalue of the discrete $$p$$-Laplacian on $$X$$. From this we deduce a very simple proof of a result by Linial-London-Rabinovich (1995) and Matousek (1997): families of expanders embed in $$L^p$$ with logarithmic distortion.

## Parallel Sessions

### Mathematics, game theory, and verification of complex computer systems

Organizer: T. Brihaye (UMONS)
• 14:00 Nicolas Markey (ENS Cachan) — Quantified CTL.

Quantified CTL extends the branching-time temporal logic CTL with quantification over the labellings of the Kripke structure with atomic propositions (with several possible semantics, whether we label the structure or its computation tree). This logic was introduced long ago, in order to increase the expressiveness of CTL, but only a few results had been established.

In this talk, I will revisit this logic with a new application in mind (namely for reasoning about strategies in games). On the expressiveness side, we prove that it is as expressive as Monadic Second Order Logic; on the algorithmic side, we prove that model checking is decidable, and that satisfiability is decidable when labellings refer to the execution tree (and undecidable when they refer to the structure). This study provides a nice and uniform approach to several problems about strategic reasoning in games, which I will sketch at the end of the talk.

This is joint work with François Laroussinie.

• 14:45 James Worrell (Université d'Oxford) — Positivity Problems for Linear Recurrence Sequences.

Given a linear recurrence sequence over the reals (such as the Fibonacci numbers), the Positivity Problem asks whether the terms of the sequence are all positive, while the Ultimate Positivity Problem asks whether all but finitely many terms of the sequence are positive. Positivity and Ultimate Positivity have applications in many different areas, such as theoretical biology (analysis of L-systems, population dynamics), software verification (termination of linear programs), probabilistic model checking, quantum automata, combinatorics, term rewriting, and the study of generating functions.

The decidability of Positivity and Ultimate Positivity for rational linear recurrence sequences is a long-standing open problem (mentioned, for example, in [1]).

Our main results are as follows. Positivity and Ultimate Positivity are decidable for linear recurrence sequences of order 5; obtaining decidability for either problem for sequences of order 6 would entail major breakthroughs in the field of Diophantine approximation of transcendental numbers; for simple linear recurrence sequences, i.e., those with no repeated characteristic roots, Positivity is decidable up to order 9 and Ultimate Positivity is decidable at all orders. These results make use of lower bounds in Diophantine approximation as well as decision procedures for real closed fields.

This is joint work with Joël Ouaknine.

[1] A. Salomaa. Growth functions of Lindenmayer systems: Some new approaches. In A. Lindenmayer and G. Rozenberg, editors, Automata, Languages, Development. North-Holland, 1976.
• 16:00 Quentin Menet (UMONS) — Banach-Mazur games.

The similarities between sets of probability $$1$$ and residual sets have always intrigued mathematicians. Surprisingly, game theory allows us to obtain a nice characterization of residual sets thanks to Banach-Mazur games [1]. We will be particularly interested by the subsets of infinite paths on finite graphs. In this context, we can in fact imagine several kind of strategies [2] for Banach-Mazur games each of which implies the residuality. We then remark that the existence of certain winning simple strategies for Banach-Mazur games also imply that the set is of probability $$1$$ for some reasonable probability measures. One can therefore wonder whether there exists a kind of strategy for Banach-Mazur games allowing us to characterize sets of probability $$1$$.

We will start by introducing Banach-Mazur games and by comparing simple strategies in the case of finite graphs. We will then study their link with the sets of probability $$1$$ and we will look at a generalization of Banach-Mazur games allowing us to characterize certain families of sets of probability $$1$$. This talk will mainly based on the paper [3].

[1] J. C. Oxtoby, The Banach-Mazur Game and Banach Category Theorem, Annals of Mathematical Studies (1957), 39, 159–163.
[2] E. Grädel and S. Leßenich, Banach-Mazur Games with Simple Winning Strategies, CSL 2012, 305–319.
[3] T. Brihaye and Q. Menet, Simple strategies for Banach-Mazur games and fairly correct systems, GandALF 2013.
• 16:45 Mickael Randour (UMONS) — Beyond Worst-Case Synthesis in Two-Player Quantitative Games.

In this work, we consider two-player games played on graphs. Such games are commonly used for formal verification: they model interactions between a reactive system (player 1) and its uncontrollable environment (player 2). The system aims to ensure some specification given as an objective of the game, despite the actions taken by the environment. We are particularly interested in automated synthesis of provably safe and efficient controllers: a strategy of player 1 that guarantees winning the game represents such a controller.

Many real-world applications require quantitative requirements to be specified (e.g., minimize the response time of a system). Two views co-exist regarding the model of the environment. On the one hand, classical analysis of two-player games involves an opponent which is purely antagonistic and asks for strict guarantees. On the other hand, stochastic models like Markov decision processes represent situations where the system is faced to a purely randomized environment: the aim is then to optimize the expected payoff, without guarantee on individual outcomes.

We introduce the beyond worst-case synthesis problem, which is to construct strategies that guarantee some quantitative requirement in the worst-case while providing an higher expected value against a particular stochastic model of the environment given as input. This problem is relevant to produce system controllers that provide nice expected performance in the everyday situation while ensuring a strict (but relaxed) performance threshold even in the event of very bad (while unlikely) circumstances.

We study the beyond worst-case synthesis problem for two important quantitative settings: the shortest path and the mean-payoff. In both cases, we show how to decide the existence of finite-memory strategies satisfying the problem and how to synthesize one if one exists. We establish algorithms and we study complexity bounds and memory requirements. Based on joint work with V. Bruyère, E. Filiot and J.-F. Raskin.

### Algebraic Topology

Organizer: D. Chataur (Lille 1)
• 14:00 Pedro Forte Vaz (UCL) — On Jaeger's HOMFLY-PT expansions, branching rules and link homology.

In 1989 François Jaeger showed that the Kauffman polynomial of a link L can be obtained as a weighted sum of HOMFLYPT polynomials of a certain family of links associated to L. In this talk I will explain how to obtain more general expansions using the so called "branching rules" for Lie algebra representations and describe how to to obtain categorified versions of Jaeger's expansions for link homology theories using categorical versions of branching rules.

• 14:45 Irakli Patchkoria (Universität Bonn) — Rigidity in equivariant stable homotopy theory.

For any finite group G, we show that the 2-local G -equivariant stable homotopy category, indexed on a complete G -universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy cofiber sequences and the stable Burnside category determine all "higher order structure" of the 2-local G-equivariant stable homotopy category, such as the equivariant homotopy types of function G-spaces. The theorem can be seen as an equivariant version of Schwede's rigidity theorem at the prime 2.

• 16:00 Lucile Vandembroucq (Universidade do Minho, Braga, Portugal) — Topological Complexity and related invariants.

The topological complexity of a space has been introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem in robotics. In this talk, I will discuss the relationships between Farber's Topological Complexity and related invariants such as Iwase-Sakai's Monoidal Topological Complexity, Doeraene-El Haouari's relative category and the LS-category of the cofibre of the diagonal map. In particular, I will present some new results obtained in collaboration with José Calcines and José Carrasquel which are related to the Iwase-Sakai conjecture (asserting that Topological Complexity coincides with Monoidal Topological Complexity) and the Doeraene- El Haouari conjecture (asserting that the relative category of a map f coincides with the sectional category when f admits a homotopy retraction).

• 16:45 Urtzi Buijs (UCL) — Rational homotopy of non-connected spaces.

Classical rational homotopy theory relies in the equivalence of the usual homotopy category of simply connected, or more generally, nilpotent rational CW-complexes with the homotopy categories of commutative differential graded algebras and differential graded Lie algebras concentrated, roughly speaking, in positive degrees. Here, we show that considering the unbounded situation is also of interest when studying the rational homotopy type of non-connected spaces.

### Applied Analysis

Organizer: T. Claeys (UCL)
• 14:00 Arno Kuijlaars (KU Leuven) — Squared singular values of products of random matrices.

The squared singular values of a random matrix with entries that are independent complex Gaussians are a classical object of study in random matrix theory. It is known that they are determinantal point process with a correlation kernel that is built out of Laguerre polynomials. In the limit of large dimension the squared singular values have a limiting distribution that is known as the Marchenko-Pastur distribution.

We discuss more recent results for the singular values of a product of such random matrices. Akemann, Ipsen and Kieburg showed that the determinantal point process is now described in terms of Meijer G-functions. It is in fact a multiple orthogonal polynomial ensemble with a new scaling limit at the origin.

[1] G. Akemann, J.R. Ipsen and M. Kieburg, Products of rectangular random matrices: singular values and progressive scattering, arXiv:1307.7560
[2] A.B.J. Kuijlaars and L. Zhang, Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits, arXiv:1308.1003
• 14:45 Mattia Cafasso (Université d'Angers) — Darboux transformations and random point processes.

In this talk we will show how the classical theory of Darboux transformations can be used to compute certain gap probabilities appearing in random matrix theory and related subjects. Examples will include the Airy process with outliers, the Pearcey process with inliers and the eigenvalue distribution of the Gaussian unitary ensemble with external potential. These results have been obtained in collaboration with Marco Bertola.

• 16:00 Maxim Derevyagin (TU Berlin, KU Leuven) — A tridiagonal approach to interpolation problems.

Many mathematical models are described by probability measures that do not necessarily have finite moments of all orders. Thus, one cannot directly use the powerful tools from the theory of orthogonal polynomials and moment problems to study these models. One way to overcome this obstacle is to consider an interpolation problem for the Cauchy transform of the measure instead of a moment problem. This way seems to be natural since the classical Hamburger moment problem is the limiting case of the Ne­van­lin­na-Pick interpolation problem in the upper half-plane (a special type of interpolation problems).

In this talk we will see that orthogonal polynomials, Jacobi matrices and Padé approximants appear in the framework of Nevanlinna-Pick problems in a similar way as they do in the setting of the Hamburger moment problem case. In other words, the key point in getting those objects is a generalization of Jacobi fractions or, equivalently, a Schur-type algorithm, which goes back at least to Wall, and was also rediscovered and studied by Delsarte and Genin.

More precisely, the main goal of the talk is to present the theory of these interpolating fractions from the classical spectral point of view. In particular, we will see how one can use Jacobi matrices and orthogonal functions to study Nevanlinna-Pick problems and we will discuss the uniqueness criteria for solutions of Nevanlinna-Pick problems.

• 16:45 Bernhard Beckermann (Lille 1) — Algebraic properties of robust Padé approximants.

It has been conjectured [2] that recently introduced so-called robust Padé approximants computed through SVD techniques do not have so-called spurious poles [3], that is, poles with a close-by zero or poles with small residuals. Such a result would have a major impact on the convergence theory of Padé approximants since it is known that convergence in capacity plus absence of poles in some domain $$D$$ implies locally uniform convergence in $$D$$.

Following [1], we prove in the present talk the conjecture for the subclass of so-called well-conditioned Padé approximants, and discuss related questions. It turns out that it is not sufficient to discuss only linear algebra properties of the underlying rectangular Toeplitz matrix, since in our results other matrices like Sylvester matrices also occur. This type of matrices have been used before in numerical greatest common divisor computations.

Joint work with Ana C. Matos (Lille).

[1] B. Beckermann and A.C. Matos, Algebraic properties of robust Padé approximants. Manuscript (2013).
[2] P. Gonnet, S. Güttel and L. N. Trefethen, Robust Padé approximation via SVD, SIAM Review, 55 (2013), pp. 101–117.
[3] H. Stahl, Spurious poles in Padé approximation, J. Comp. Appl. Math., 99 (1998), 511–527.

### Functional Analysis

Organizers: D. Li (Université d'Artois) & K. Grosse-Erdmann (UMONS)

On Tuesday:

• 14:00 Gilles Godefroy (Jussieu) — Remarks on the uniqueness of isometric preduals.

Two very different Banach spaces may have isometric dual spaces: a significant example is provided by the large collection of isometric preduals of $$l_1$$. However, simple conditions on the space or the dual space suffice to show the uniqueness of preduals. We will present some simple recent results on this topic, based e.g. on elementary Baire category arguments. We will also recall a bunch of natural open questions in the field. Technicalities will be avoided.

• 14:45 Emmanuel Fricain (Lille 1) — A brief review on Carleson type embeddings.

Let $$H^2$$ be the Hardy space of the unit disc $$\mathbb D$$. A famous theorem of Carleson characterizes positive and finite Borel measures $$\mu$$ on the unit disc such that the space $$H^2$$ embeds continuously into $$L^2(\mu)$$. In this talk, we briefly review some results of this type for some other spaces of analytic functions on $$\mathbb D$$. In particular, we pay attention to the case of the de Branges–Rovnyak spaces. We also discuss some recent reversed embeddings results.

A part of this talk is based on joint works with A. Baranov, A. Blandignères, F. Gaunard, A. Hartmann, J. Mashreghi and W. Ross.

• 16:00 Stefaan Vaes (KU Leuven) — Uniqueness of Cartan subalgebras in II₁ factors and type III factors.

The group measure space construction of Murray and von Neumann associates to every free and ergodic action of a countable group $$\Gamma$$ on a measure space $$(X,\mu)$$, the crossed product von Neumann algebra $$M = L^\infty(X) \rtimes \Gamma$$, which can be of type I, II or III. The subalgebra $$L^\infty(X)$$ of $$M$$ is a Cartan subalgebra. In order to classify these crossed product von Neumann algebras $$M$$ in terms of the group action $$\Gamma \curvearrowright (X,\mu)$$, the first problem is to decide whether $$L^\infty(X)$$ is the unique Cartan subalgebra of $$M$$. I will give a survey of the recent advances on this question, both in the type II and in the type III case, focusing on  [3,4,1,2].

[1] C. Houdayer and S. Vaes, Type III factors with a unique Cartan decomposition. To appear in J. Math. Pures Appl. arXiv:1203.1254
[2] A. Ioana, Cartan subalgebras of amalgamated free product II₁ factors. Preprint. arXiv:1207.0054
[3] S. Popa and S. Vaes, Unique Cartan decomposition for II₁ factors arising from arbitrary actions of free groups. To appear in Acta Math. arXiv:1111.6951
[4] S. Popa and S. Vaes, Unique Cartan decomposition for II₁ factors arising from arbitrary actions of hyperbolic groups. To appear in J. Reine Angew. Math. arXiv:1201.2824

On Thursday:

• 14:00 Philippe Jaming (Université de Bordeaux 1) — Heisenberg Uniqueness Pairs.

Let $$C$$ be a smooth curve in the plane and $$A$$ be a set of lines in the plane. $$(S; A)$$ is a Heisenberg uniqueness pair (HUP) if the only finnite measure $$m$$ that is absolutely continuous with respect to arc length on $$C$$ and such that its Fourier transform vanishes on $$A$$ is $$m = 0$$. In this talk we will show how this notion can be reformulated in geometric terms. This allows us to extend results of Hendelmalm, Sjolin and Lev to rather general curves.

This is joint work with Karim Kellay.

• 14:45 Quentin Menet (UMONS) — Hypercyclic subspaces.

Linear dynamics studies the behaviour of linear dynamical systems, that is, the properties of orbits of operators on Banach or Fréchet spaces. This young theory has rapidly evolved over the past two decades and is now the subject of two great books [1,3].

A key notion of linear dynamics is the notion of hypercyclic operators. An operator $$T$$ on a Fréchet space $$X$$ is said to be hypercyclic if there is a vector $$x$$ in $$X$$ (also called hypercyclic) whose the orbit under $$T$$ is dense. After a state-of-the-art of the main notions in linear dynamics, we will focus on the notion of hypercyclic subspaces. We say that $$T$$ possesses a hypercyclic subspace if there exists an infinite-dimensional closed subspace in which every non-zero vector is hypercyclic. In 2000, a characterization of operators with hypercyclic subspaces was obtained by González, León and Montes [2] in the case of complex Banach spaces by using spectral theory. However, so far no characterization of operators with hypercyclic subspaces on Fréchet spaces is known. In the talk we will present some recent advances concerning this interesting open problem.

[1] F. Bayart and É. Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, Cambridge University Press, 2009.
[2] M. González, F. León-Saavedra and A. Montes-Rodríguez, Semi-Fredholm theory: hypercyclic and supercyclic subspaces, Proc. London Math. Soc. (3) 81 (2000), 169–189.
[3] K.-G. Grosse-Erdmann and A. Peris, Linear chaos, Springer, London, 2011.
• 16:00 Samuel Nicolay (ULg) — Generalized pointwise Hölder spaces.

In [8,7], the properties of generalized uniform Hölder spaces have been investigated. The idea underlying the definition is to replace the exponent $$\alpha$$ of the usual spaces $$\Lambda^\alpha(\mathbb{R}^d)$$ (see e.g. [6]) with a sequence $$\sigma$$ satisfying some conditions. The so-obtained spaces $$\Lambda^\sigma(\mathbb{R}^d)$$ generalize the spaces $$\Lambda^\alpha(\mathbb{R}^d)$$; the spaces $$\Lambda^\sigma(\mathbb{R}^d)$$ are actually the spaces $$B^{1/\sigma}_{\infty,\infty}(\mathbb{R}^d)$$, but they present specific properties (induced by $$L^\infty$$-norms) when compared to the more general spaces $$B^{1/\sigma}_{p,q}(\mathbb{R}^d)$$ studied in  [2,4,1,5,9,10] for example. Indeed it is shown in [8,7] that most of the usual properties holding for the spaces $$\Lambda^\alpha(\mathbb{R}^d)$$ can be transposed to the spaces $$\Lambda^\sigma(\mathbb{R}^d)$$.

Here, we introduce the pointwise version of these spaces: the spaces $$\Lambda^{\sigma,M}(x_0)$$, with $$x_0 \in\mathbb{R}^d$$. Let us recall that a function $$f\in L^\infty_\text{loc}(\mathbb{R}^d)$$ belongs to the usual pointwise Hölder space $$\Lambda^\alpha(x_0)$$ ($$\alpha>0$$) if and only if there exist $$C,J>0$$ and a polynomial $$P$$ of degree at most $$\alpha$$ such that $\sup_{|h|\le 2^{-j}} |f(x_0+h)-P(h)| \le C 2^{-j\alpha}.$ As in [8,7], the idea is again to replace the sequence $$(2^{-j\alpha})_j$$ appearing in this inequality with a positive sequence $$(\sigma_j)_j$$ such that $$\sigma_{j+1}/\sigma_j$$ and $$\sigma_j/\sigma_{j+1}$$ are bounded (for any $$j$$); the number $$M$$ stands for the maximal degree of the polynomial (this degree can not be induced by a sequence $$\sigma$$). By doing so, one tries to get a better characterization of the regularity of the studied function $$f$$. Generalizations of the pointwise Hölder spaces have already been proposed (see e.g. [3]), but, to our knowledge, the definition we give here is the most general version and leads to the sharpest results.

[1] Alexandre Almeida. Wavelet bases in generalized Besov spaces. J. Math. Anal. Appl., 304(1):198–211, 2005.
[2] António M. Caetano and Susana D. Moura. Local growth envelopes of spaces of generalized smoothness: the critical case. Math. Inequal. Appl., 7(4):573–606, 2004.
[3] Marianne Clausel. Quelques notions d'irrégularité uniforme et ponctuelle : le point de vue ondelettes. PhD thesis, University of Paris XII, 2008.
[4] Walter Farkas. Function spaces of generalised smoothness and pseudo-differential operators associated to a continuous negative definite function. Habilitation Thesis, 2002.
[5] Walter Farkas and Hans-Gerd Leopold. Characterisations of function spaces of generalised smoothness. Ann. Mat. Pura Appl., IV. Ser., 185(1):1–62, 2006.
[6] Steven G. Krantz. Lipschitz spaces, smoothness of functions, and approximation theory. Exposition. Math., 1(3):193–260, 1983.
[7] Damien Kreit and Samuel Nicolay. Characterizations of the elements of generalized Hölder-Zygmund spaces by means of their representation. J. Approx. Theory, to appear, 10.1016/j.jat.2013.04.003.
[8] Damien Kreit and Samuel Nicolay. Some characterizations of generalized Hölder spaces. Math. Nachr., 285(17-18):2157–2172, 2012.
[9] Thomas Kühn, Hans-Gerd Leopold, Winfried Sickel, and Leszek Skrzypczak. Entropy numbers of embeddings of weighted Besov spaces II. Proceedings of the Edinburgh Mathematical Society (Series 2), 49(02):331–359, 2006.
[10] Susana D. Moura. On some characterizations of Besov spaces of generalized smoothness. Math. Nachr., 280(9-10):1190–1199, 2007.

### Algebra and applications

Organizer: A. Leroy (Université d'Artois)

On Tuesday:

• 14:00 Jacques Alev (Université de Reims) — The noetherian property in noncommutative algebra.

Starting with the basic question of the noetherianity of the enveloping algebra of the Virasoro algebra, we will see how the non commutative analogs of certain standard non noetherian commutative rings tend to be noetherian.

• 14:45 Antoine Touzé (Paris XIII) — Schur algebras, functors and homological computations.

Friedlander and Suslin showed that the category of representations of the Schur algebra $$S(n,d)$$ is equivalent (when $$n$$ is greater than $$d$$) to the so called category of strict polynomial functor $$P_d$$.

In this talk, we will briefly describe the category $$P_d$$, and give some applications of this point of view to homological algebra computations.

• 16:00 Agata Smoktunowicz (University of Edinburgh) — Some open questions in noncommutative ring theory.

There are many longstanding open questions in noncommutative ring theory, which are easy to formulate and easily understood. One such problem is very well known, namely the Kurosh conjecture on domains. This asks whether a finitely generated algebraic algebra which is a domain is finite dimensional. A related open question (Latyshev, 1970's) asks whether there exists a finitely generated ring which is infinite and which is also a division ring. Other basic open questions include the Koethe conjecture on nil rings (1930), and the question whether finitely presented nil algebras are nilpotent. We will look at some methods which are used in this area, for example the Golod-Shafarevich theorem, as well as some partial results which are known to be related to these questions. We will also look at some related new and old results on algebraic algebras and free algebras, Golod-Shafarevich algebras, domains and Noetherian algebras, growth of algebras and the Gelfand-Kirillov dimension. Connections between noncommutative ring theory, group theory and noncommutative (projective) algebraic geometry and other areas of mathematics will also be mentioned.

On Thursday:

• 14:00 Jerzy Matczuk (University of Warsaw, Varsovie) — Idempotents and Clean Elements in Ring Extensions.

The aim of the talk is to present some results on idempotents and clean elements of ring extensions $$R\subseteq S$$ where $$S$$ stands for one of the rings $$R[x_1,x_2,\dots,x_n]$$, $$R[x_1^{\pm 1},x_2^{\pm 1},\dots,x_n^{\pm 1}]$$, $$R[[x_1,x_2,\dots,x_n]]$$.

In particular, criterions for an idempotent of $$S$$ to be conjugate to an idempotent of $$R$$ will be presented. As an application we show that all idempotents of the power series ring are conjugate to idempotents of the base ring and we apply this to get a new proof of the result of P.M. Cohn that the ring of power series over a projective-free ring is also projective-free.

It is well known that the polynomial ring is never a clean ring. We present a description of the set of all clean elements in the polynomial $$R[x]$$ over a 2-primal ring (i.e. $$R/B(R)$$ is a reduced ring, where $$B(R)$$ is the prime radical of $$R$$). It appears that, if in addition $$R$$ is a clean ring, then the set $$Cl(R[x])$$ of all clean elements of $$R[x]$$ forms a subring of $$R[x]$$. We show also that the Koethe's conjecture has a positive solution if and only if for any clean ring $$R$$ such that $$R/\text{Nil}(R)$$ is a reduced ring, the set $$Cl(R[x])$$ forms a subring of $$R[x]$$.

(The talk is based on a joint work with P. Kanwar and A.Leroy.)

• 14:45 Jean Pierre Tignol (UCL) — Valuation theory for algebras with involution.

Valuation theory plays a central role in the solution of various problems concerning finite-dimensional division algebras, such as the construction of noncrossed products and of counterexamples to the Kneser-Tits conjecture. However, relating valuations to Brauer-group properties is particularly difficult because valuations are defined only on division algebras and not on central simple algebras with zero divisors. This talk will present a more flexible tool recently developed in a joint work with Adrian Wadsworth, which applies to a broad spectrum of noncommutative situations. In particular, central simple algebras with anisotropic involution over Henselian fields are shown to carry a special kind of value function, which is an analogue of Schilling valuations on division algebras.

• 16:00 Dimitri Gourevitch (UVHC) — A new approach in Noncommutative Algebra and its applications to Mathematical Physics.

Let $$A$$ be a Noncommutative algebra. One of the basic questions of Noncommutative Algebra and Geometry –what is a natural analog of differential algebra on $$A$$–has no evident answer. There are known a few approaches to this problem. They will be mentioned in our talk. Recently we succeeded in constructing a differential algebra on the enveloping algebra $$U(gl(n))$$, which is a deformation of the usual one on the symmetric commutative algebra $$Sym(gl(n))$$. Our approach gives rise to a new version of Noncommutative space-time. Surprisingly, this entails the discreteness of this space-time. The role of the so-called Braided Algebra in our approach will be explained.

### Numerical Schemes for PDEs

Organizers: E. Creusé (Lille 1) & S. Nicaise (UVHC)
• 14:00 Franck Boyer (Université Aix-Marseille) — Analysis of the upwind finite volume scheme for non smooth initial and boundary value transport problem.

This talk, based on reference [2], is concerned with the analysis of the upwind finite volume scheme on general grids for the linear transport problem with as few as possible regularity assumption on the data. More precisely, we consider bounded solutions and velocity fields with Sobolev regularity in space, following the framework of renormalized solutions introduced by DiPerna and Lions.

Our main result is the uniform in time strong convergence of the approximate solutions with values in $$L^p(\Omega)$$ for any $$p<+\infty$$, as well as the strong convergence of the traces in appropriate spaces.

In the first part of the talk, I will recap the main elements of the renormalised solutions theory [4], including its extension to the trace problem [1,3], which is used in the present work.

[1] F. Boyer. Trace theorems and spatial continuity properties for the solutions of the transport equation. Differential Integral Equations, 18(8):891–934, 2005.
[2] F. Boyer. Analysis of the upwind finite volume method for general initial- and boundary-value transport problems. IMA J. Numer. Anal., 32(4):1404–1439, 2012.
[3] F. Boyer and P. Fabrie. Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, volume 183 of Applied Mathematical Sciences. Springer, New York, 2013.
[4] R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98(3):511–547, 1989.
• 14:45 Patrick Dular (ULg) — Subproblem finite element method for magnetostatic and magnetodynamic model refinements.

Model refinements of magnetostatic and magnetodynamic problems are performed via a subproblem finite element method. A complete problem is split into subproblems with overlapping meshes, to allow a progression from source to reaction fields, ideal to real flux tubes, 1-D to 2-D to 3-D models, perfect to real materials, linear to nonlinear materials, thin shell to volume models, statics to dynamics, with any coupling of these changes. Its solution is the sum of the subproblem solutions. The procedure simplifies both meshing and solving processes, and quantifies the gain given by each refinement on both local fields and global quantities. The developments are performed paying special attention to the proper discretization of the constraints involved in each subproblem. Efficient ways to chain the refinements are discussed with references to common problems.

• 16:00 Martin Vohralík (Inria Paris-Rocquencourt, Projet Pompadi) — Adaptive inexact Newton methods with a posteriori stopping criteria and applications.

We consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations. To solve these systems, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. We derive adaptive stopping criteria for both iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error significantly. Our estimates also yield a guaranteed upper bound on the overall error at each step of the nonlinear and linear solvers. We prove the (local) efficiency and robustness of the estimates with respect to the size of the nonlinearity owing, in particular, to the error measure involving the dual norm of the residual. Our developments hinge on equilibrated flux reconstructions and yield a general framework. We show how to apply this framework to various discretization schemes like finite elements, nonconforming finite elements, discontinuous Galerkin, finite volumes, and mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers. Numerical experiments for the $$p$$-Laplacian illustrate the tight overall error control and important computational savings achieved in our approach; an example of application to an unsteady degenerate system of partial differential equations from subsurface modeling is also presented. More details, analysis, and results can be found in [1,2], while the subsurface application is described in [3].

[1] A. Ern and M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. 35 (4), A1761–A1791, 2013.
[2] A. Ern and M. Vohralík, Adaptive inexact Newton methods: a posteriori error control and speedup of calculations, SIAM News 46 (1), 1,4, 2013.
[3] M. Vohralík and M. F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows. Comput. Geosci., DOI 10.1007/s10596-013-9356-0, 2013.

### Submanifold Geometry

Organizer: L. Vrancken (UVHC)
• 14:00 Jason Lotay (University College London, England) — Coassociative conifolds.

Coassociative 4-folds are calibrated submanifolds in 7-ma­ni­folds with holonomy group contained in $$\mathrm{G}_2$$. Of especial importance are coassociative conifolds, which have asymptotically conical ends or conical singularities. Key questions for conifolds are intimately related to the spectrum of an elliptic operator on the cross-section of the cone. From this spectrum, based on the pioneering work of Joyce, one is led to define an integer invariant associated with the cone called the stability index. I will describe connections that the stability index has to various problems for coassociative conifolds including: deformation theory, gluing problems and existence and uniqueness questions.

• 14:45 Henri Anciaux (Universidade de São Paulo, Brazil) — Surfaces with one constant principal curvature in real space forms and Hopf hypersurfaces in complex space forms.

We shall report on two related works. The first one (available at arXiv:1307.6735) establishes the local classification of surfaces having one constant principal curvature in the six pseudo-Riemannian $$3$$-dimensional space forms, namely: the Euclidean space $$\mathbb E^3$$, the Minkowski space $$\mathbb L^3$$, the $$3$$-sphere $$\mathbb S^3$$, the hyperbolic space $$\mathbb H^3$$, the de Sitter and anti de Sitter spaces $$d\mathbb S^3$$ and $$Ad\mathbb S^3$$. Such surfaces are characterized as being "generalized" tubes over curves, i.e. the set of points which are "equidistant" (in some sense) to a regular curve. This generalizes and unifies previous results of Shiohama/Tagaki (in Euclidean space) and Aledo/Gálvez (in hyperbolic space).

The second work, in progress, is a collaboration with K. Panagiotidou (Aristotle University of Thessaloniki). It deals with Hopf hypersurfaces of complex or para-complex pseudo-Riemannian space forms (thus including the complex projective space $$\mathbb C \mathbb P^n$$ and the complex hyperbolic space $$\mathbb C \mathbb H^n$$), namely those hypersurfaces whose Hopf field is a principal direction. The situation is similar to the previous one since this assumption implies the constancy of the corresponding principal curvature. Again we generalize and unify several local classification results du to Cecil/Ryan, Montiel and Ivey, proving that Hopf hypersurfaces are, locally, "generalized" tubes over complex submanifolds.

• 16:00 Joeri Van der Veken (KU Leuven) — Inequalities for Lagrangian submanifolds.

This talk is based on joint work with Bang-Yen Chen, Franki Dillen and Luc Vrancken on curvature inequalities for Lagrangian submanifolds of complex space forms. In particular, if $$M^n$$ is an $$n$$-dimen­sional Lagrangian submanifold of a complex space form $$\tilde M^n(c)$$ of constant holomorphic sectional curvature $$c$$, then one can prove pointwise inequalities of the following type: $d(n_1,\ldots,n_k) \leq a(n,k,n_1,\ldots,n_k) \|H\|^2 + b(n,k,n_1,\ldots,n_k) c.$ Here, $$H$$ is the mean curvature vector and $$d(n_1,\ldots,n_k)$$ is any delta-curvature of $$M^n$$. Recall that on an $$n$$-dimensional Riemannian manifold, one can define a delta-curvature for any set $$\{n_1,\ldots,n_k\}$$ of integers, satisfying $$2 \leq n_1 \leq \cdots \leq n_k \leq n-1$$ and $$n_1 + \cdots + n_k \leq n$$. The strength of the inequalities lies in the fact that they give information about intrinsic invariants (the delta-curvatures) by knowing an extrinsic invariant (the mean curvature) and vice versa.

A first proposal for the coefficients $$a$$ and $$b$$ in the inequality was given in [1]. However, if, at some point of the Lagrangian submanifold, equality was attained in the resulting inequality, the mean curvature had to vanish at that point. This suggested that the coefficient $$a$$ could be improved. Indeed, in  [2] a sharper value for $$a$$ was given. However, the new inequality was only proven under some extra assumption on the integers $$n_1,\ldots,n_k$$. Finally, in [3], we gave a complete proof of the inequality proposed in [2] in all cases and we sharpened the inequality even further in the special case that $$n_1+\cdots+n_k=n$$. In both cases, there are examples of submanifolds realizing equality at every point, without being minimal. This shows that the inequalities are now optimal.

[1] B.-Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Totally real submanifolds of $$CP^n$$ satisfying a basic equality, Arch. Math. 63 (1994), 553–564.
[2] B.-Y. Chen and F. Dillen, Optimal general inequalities for Lagrangian submanifolds in complex space forms, J. Math. Anal. Appl. 379 (2011), 229–239.
[3] B.-Y. Chen, F. Dillen, J. Van der Veken and L. Vrancken, Curvature inequalities for Lagrangian submanifolds: the final solution, preprint.
• 16:45 Olivier Birembaux (UVHC) — Isotropic affine hypersurfaces of dimension 5.

Note that we say that a bundle valued tensor $$T$$ is isotropic if and only if the value of $$g(T(v,\dots,v),T(v,\dots,v))$$ is independent of the unit vector $$v$$. If we take for $$T$$ the second fundamental form of a submanifold of a real space form, these submanifolds can be considered as a natural genera­lization of the totally geodesic submanifolds.

In the affine differential geometry, there is a natural tensor, called the difference tensor which is defined as the difference between the induced affine connection and the Levi Civita connection of the affine metric. The classical Pick-Berwald theorem states that the difference tensor vanishes if and only if $$M$$ is a nondegenerate quadric. In that sense this tensor corresponds to the second fundamental form for submanifolds of real space forms.

Here we will study affine hypersurfaces for which the difference tensor is isotropic, i.e. we will assume that $h(K(v,v),K(v,v))=\lambda^2(p) (h(v,v))^2,$ for any tangent vector $$v$$ at any point $$p$$.

Note that any surface always has isotropic difference tensor. Therefore, we may assume that $$n >2$$. Such hypersurfaces have previously been studied by O. Birembaux and Mirjana Djorić under the additional assumption that $$M$$ is an affine hypersphere.

Here we study the general case. As for affine spheres, we first show that isotropic affine hypersurfaces which are not congruent to quadrics are necessarily $$5$$, $$8$$, $$14$$ or $$26$$ dimensional. From this, we also obtain a complete classification in dimension 5.

### Theory and applications of categories

Organizer: I. Stubbe (ULCO)
• 14:00 Steve Lack (Macquarie University, Sydney) — The Catalan simplicial set.

The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. By looking at these families of sets, rather than the mere sequence of natural numbers, one can see various types of extra structure. For example Tamari has shown that each set in the family can be given the structure of a lattice.

In this talk I shall explain how the family itself can be given the structure of a simplicial set. Simplicial sets are purely combinatorial structures, but are most often used in the context of homotopy theory; and, more recently, in higher category theory. I shall also explain the significance of the Catalan simplicial set in higher category theory. The low-dimensional parts of the simplicial set classify, in a precise sense, the structures of monoid and of monoidal category, and it appears that the full simplicial set may likewise classifies higher-dimensional analogues of these.

This work lies at the interface of combinatorics, algebraic topology, quantum groups, logic, and category theory.

• 14:45 Tim Van der Linden (UCL) — What is a tensor product internally?.

The concept of tensor product is of prime importance in homological algebra and beyond. When it is studied from a categorical perspective, it is usually treated either as additional structure on a category — which leads to the theory of monoidal and enriched categories — or in an ad-hoc way involving free algebras of some kind. Quite surprisingly, as yet no internal categorical construction in terms of limits and colimits has been proposed.

The aim of my talk is to do precisely this. I will first state a general construction of such an intrinsic tensor product, based on the work [1,2,3] in the context of semi-abelian categories [4]. Then I will give an overview of the main examples and sketch some applications.

This is joint work with Manfred Hartl.

[1] A. Carboni and G. Janelidze, Smash product of pointed objects in lextensive categories, J. Pure Appl. Algebra 183 (2003), 27–43.
[2] M. Hartl and B. Loiseau, On actions and strict actions in homological categories, Theory Appl. Categ. 27 (2013), no. 15, 347–392.
[3] M. Hartl and T. Van der Linden, The ternary commutator obstruction for internal crossed modules, Adv. Math. 232 (2013), no. 1, 571–607.
[4] G. Janelidze, L. Márki, and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), no. 2–3, 367–386.
• 16:00 Joost Vercruysse (VUB) — Quasi-Frobenius Functors.

A functor that has the same left and right adjoint is called a Frobenius functor. This name is inspired by the fact that the forgetful functor from the category of A-modules to k-modules possesses this property if and only if A is a Frobenius k-algebra. To capture quasi-Frobenius algebras by a similar property, we introduced the notion of a quasi-Frobenius functor F between Abelian categories as a functor that has both a left adjoint L and a right adjoint R, such that are L and R similar. In this talk we will discuss the basic properties of these functors and the relation with the quasi-Frobenius rings, bimodules and corings. This is based on joint work with F. Castano-Iglesias and C. Nastascescu.

• 16:45 Tom Leinster (University of Edinburgh) — The eventual image.

An endomorphism $$T$$ of an object can be viewed as a discrete-time dynamical system: perform one iteration of $$T$$ with every tick of the clock. This dynamical viewpoint suggests questions about the long-term destiny of the points of our object. (For example, does every point eventually settle into a periodic cycle?)

A fundamental concept here is the eventual image, a companion to the algebraic concepts of image and kernel. Under suitable hypotheses, it can be defined as the intersection of the images of all the iterates $$T^n$$ of $$T$$. I will explain its behaviour in three settings: one set-theoretic, one algebraic, and one geometric. I will also present a unifying categorical framework, showing that the eventual image has not one but two universal properties, dual to one another. In this, it resembles other important constructions such as direct sum in an abelian category.

### Nonlinear elliptic PDEs

Organizers: C. De Coster (UVHC) & C. Troestler (UMONS)
• 14:00 Denis Bonheure (ULB) — Born Infeld equations with sources.

I will review some of the challenges from the nonlinear theory of electromagnetism formulated by Born and Infeld and I will present some recent partial results.

• 14:45 Guido Sweers (Universität zu Köln) — The clamped plate under uniform load.

It is well known that the Dirichlet bilaplace boundary value problem, which is used as a model for the clamped plate, is not sign-preserving in general. Sign-preserving means that a positive right hand side $$f$$ implies that the solution of $$\Delta^2 u =f$$ in $$\Omega$$ with $$u=\left|\nabla u\right|=0$$ on $$\partial\Omega$$ is positive. It is also known that the corresponding first eigenfunction may be sign-changing. But what will happen for a uniform load, i.e. $$f=1$$? Will the solution be positive?

This is a joint work with Hans-Christoph Grunau.

• 16:00 Louis Jeanjean (Université de Franche-Comté) — Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations.

The aim of this talk is to study the existence and the stability of standing waves with prescribed $$L^2$$-norm for a class of Schrödinger-Poisson-Slater equations in $$\mathbb R^{3}$$ $$\label{evolution1} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0$$ when $$p \in (\frac{10}{3},6)$$. The standing waves are found as critical points of the associated energy functional \begin{equation*} F(u) = \frac{1}{2}\int_{{\mathbb R}^3}|\nabla u|^2 \mathrm{d}x + \frac{1}{4} \int_{{\mathbb R}^3} \hspace{-2pt} \int_{{\mathbb R}^3} \frac{|u(x)|^2 |u(y)|^2}{|x-y|} \mathrm{d}x \mathrm{d}y - \frac{1}{p} \int_{{\mathbb R}^3}|u|^p \mathrm{d}x \end{equation*} on the constraints given by $S(c) = \{u\in H^{1}(\mathbb R^{3}): \|u\|_{2}^2=c\}$ where $$c>0$$ is given. For the value of $$p \in (\frac{10}{3},6)$$ considered the functional $$F$$ is unbounded from below on $$S(c)$$ and the existence of critical points is obtained by a mountain pass theorem on $$S(c)$$. In order to show the compactness of the Palais-Smale sequences, we prove the monotonicity of the mountain pass energy levels $$\gamma(c)$$ as well as a localization lemma for a specific sequence. Our main result is that standing waves with prescribed $$L^2$$-norm exist provided that $$c>0$$ is sufficiently small. We shall also see that when $$c>0$$ is not small a non-existence result is expected. The solutions obtained are shown to be strongly orbitaly unstable. Finally we draw a comparison between the Schrödinger-Poisson-Slater equation and the classical nonlinear Schrödinger equation.

This talk mainly describes joint works with Jacopo Bellazzini (Sassari-Italy) and Tingjian Luo (Univ. Franche-Comté) [1,2].

[1] J. Bellazzini, L. Jeanjean, T. Luo, Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. London Math. Soc., 107, 2013, 303-339.
[2] L. Jeanjean, T. Luo, Sharp non-existence results of prescribed $$L^2$$-norm solutions for some class of Schrödinger-Poisson and quasilinear equations, Zeitschrift fur Angewandle Mathematik und Physik, 64, 2013, 937-954.
• 16:45 Wolfgang Reichel (KIT) — Ground states for a nonlinear Schrödinger equation with interface.

We are interested in ground states for the nonlinear Schrö­din­ger equation (NLS) $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1} u \mbox{ in } {\mathbb R}^n,$ where the coefficients $$V$$ and $$\Gamma$$ model an interface between two purely periodic media. This means that, e.g., $$V=V_1$$ in the left halfspace $$\{x_1<0\}$$ and $$V=V_2$$ in the right halfspace $$\{x_1>0\}$$ where $$V_1, V_2$$ are periodic functions (the same applies to $$\Gamma$$). The resulting problem no longer has a periodic structure. We say that it models an interface between to periodic media.

We discuss both the case where the linear operator is positive definite or indefinite. Using variational methods and dual variational methods we give conditions on the coefficients such that ground states are supported by an interface. We also discuss example cases of explicit coefficients, where these conditions can be verified either via analytic methods or with computer assisted methods.

This is joint work with Hans-Jürgen Freisinger (KIT).

### Evolution PDEs and Applications

Organizers: C. De Coster (UVHC) & S. Nicaise (UVHC)
• 14:00 Joachim von Below (ULCO) — Stability of stationary solutions of reaction-diffusion equations on graphs.

The occurence and nonexistence of stable stationary nonconstant solutions of reaction–diffusion–equations $$\partial_t u_j=\partial_j^2 u_{j} +f_j(u_j)$$ on the edges of a finite (topological) graph are investigated under continuity and consistent Kirchhoff flow conditions at all vertices of the graph. In particular, it is shown that in the balanced autonomous case $$f(u)=u- u^3$$, no such stable stationary solution can exist if the graph is analytic. Under more general, eventually inconsistent Kirchhoff conditions, there are no stable stationary nonconstant solutions on finite trees if the derivatives of the nonlinearities are uniformly bounded from above.

• 14:45 Ali Wehbe (Université Libanaise) — Stability results of a system of coupled wave equations with one boundary dissipation law.

In this work, we study the indirect boundary stabilization of a system of coupled wave equations. We show that, the behavior of the system is greatly influenced by the algebraic property of the coupled parameter. Moreover, the energy decay rate is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations. To be more precise, under the equal speed wave propagation condition, we establish the exponential stability of the system. On the contrary, if the square root of the speed is a rational number, we show that the decay rate is polynomial.

• 16:00 Lech Zielinski (ULCO) — On the short-wave model for the Camassa-Holm equation.

The Camassa-Holm equation is a model of one-dimensional wave propagation on the free surface of shallow water that takes the form $$u_{txx} + 2u_x u_{xx} + u u_{xxx} = 2u_x$$ in the short-wave limit. The purpose of this talk is to present some results about the asymptotic behaviour of solutions obtained by an inverse scattering method.

## Poster Session

• Zainab Abbas (UVHC) — Polynomial decay rate for a wave equation with a general acoustic boundary condition.

In this poster we investigate the stability of the solution of a one dimensional wave equation with a Dirichlet boundary condition at one end and a dynamical boundary condition, generalizing the well known acoustic boundary condition, at the other.

First, we introduce the considered system and prove the existence and uniqueness of its solution.

Second, we study the behavior of the spectrum of both the dissipative system and the associated conservative system to establish an observability inequality, from which we deduce the polynomial stability. We further investigate the optimality of the decay. In order to illustrate the results, we finally present some examples.

• Elisabete Alberdi Celaya (University of the Basque Country) — Dissipation of high-frequency modes using multistep methods.

The second order Ordinary Differential Equation (ODE) system obtained after semidiscretizing the wave-type partial differential equation (PDE) with the finite element method (FEM) shows strong numerical stiffness. Its resolution requires the use of numerical methods with good stability properties and controlled numerical dissipation in the high-frequency range. Some of the methods developed in the linear range, such as the Collocation method, the Wilson method, the HHT-$$\alpha$$ method, the Houbolt method, or the generalized-alfa method are commonly used. The HHT-$$\alpha$$ method is second order precision, unconditionally stable and able to dissipate high-modes for some values of the parameters. The Backward Differentiation Formulae of second order (BDF2) is an unconditionally stable linear multistep method useful to solve ODEs of order 1.

The finite element method has been applied to the one-dimen­si­onal linear wave-type PDE and to a non-linear version of a string of a guitar. The ODE systems obtained after applying FEM are solved by the HHT-$$\alpha$$ and the BDF2, proving that both are able to dissipate the high-modes.

[1] J. Chung, G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-$$\alpha$$ method, Journal of Applied Mechanics, 60, 371-375, (1993).
[2] C. W. Gear, Numerical initial value problems in Ordinary Differential Equations, Prentice Hall, New Jersey, 1971.
[3] Ian Gladwell, Ruth Thomas, Stability properties of the Newmark, Houbolt and Wilson $$\theta$$ methods, Int. J. Numer. Anal. Methods Geomech., 4 (1980), 143-158.
[4] Hans M. Hilber, Thomas J. R. Hughes, Robert L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthq. Eng. Struct. Dyn., 5 (1977), 283-292.
[5] Hans M. Hilber, Thomas J.R. Hughes, Collocation, dissipation and overshoot for time integration schemes in structural dynamics, Earthq. Eng. Struct. Dyn., 6 (1978), 99-117.
[6] E.L. Wilson, A computer program for the dynamic stress analysis of underground structures, SEL Report, 68:1 (1968).
• Sadjiya Ariche (UVHC) — Regularity results for elliptic problems with measure.

In this work, we study the solution of the Laplace equation: $-\Delta u =g \delta_\sigma\;\;\;\hbox{ in } Q\subseteq \mathbb{R}^3,$ where $$\delta_\sigma$$ is the Dirac mass on a crack $$\sigma$$ of $$Q$$ and $$g\in L^2(\sigma)$$.

First, we discuss the existence and the uniqueness of a solution in $$W^{1,p}(Q)$$ for $$p<2$$ (due to the Dirac mass, the right-hand side is not in $$H^{-1}(Q)$$). Then, we prove the regularity of the solution and a priori estimates in weighted Sobolev spaces.

• Maya Bassam (Lebanese University) — Polynomial stability of the Timoshenko system by one boundary damping.

In this work, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. The system consists of two wave equations coupled in certain pattern. We prove that the system is strongly stable if and only if the coupling parameter is outside a discrete set of exceptional values. Moreover, we show that the energy decay rate of the system is influenced by the arithmetic property of the ratio of the wave propagation speeds of the two equations. To be more precise, under the equal speed propagation condition and if the coupling parameter is outside discrete set of exceptional values, we prove non uniform stability, however an optimal polynomial energy decay rate of type $$\frac{1}{t}$$ is obtained. On the other hand, if the ratio of the wave propagation speeds is a rational number and the coupling parameter is outside another discrete set of exceptional values, we show another polynomial energy decay rate of type $$1/{{\sqrt[3]t}}$$.

• Isabel Coelho (ULB) — Mean Curvature Problems in Unbounded Domains.

We discuss different types of problems in the context of the mean curvature equation in the Minkowski space \begin{equation*} \left(u'/{\sqrt{1 - u'^2}} \right)' = H(t,u(t)). \end{equation*}

In particular, using variational methods and a rearrangement technique, we prove the existence and one-dimensionality of heteroclinic transitions in cylinders when $$H$$ is a bistable nonlinearity. Using a shooting argument, we obtain an existence result for ground state solutions when $$H$$ is a monostable nonlinearity. Finally, when the right-hand side contains a friction term, this equation is associated with travelling waves for a reaction-diffusion problem. In this case, we present a characterisation of the solutions and the admissible wave speeds that generalise classical and recent results.

• Alexandra Convent (UCL) — Intrinsic weak derivatives and Sobolev spaces between manifolds.

Given two differentiable Riemannian manifolds $$M$$ and $$N$$ and $$p \ge 1$$, we propose an intrinsic definition of Sobolev space $$W^{1,p}(M,N)$$ without embedding the target into Euclidean spaces. In the presence of a differentiable structure, we first define colocally weakly differentiable maps and their colocal weak derivative. We then define Sobolev spaces between manifolds. Our definition is equivalent with the classical definition when $$N$$ is isometrically embedded in some Euclidean space and with the definition of Sobolev spaces into metric spaces. Finally, we endow $$W^{1,p}(M,N)$$ with a natural metrizable topology induced naturally by composition. If $$N$$ is complete, we also study distances on the Sobolev class that make it complete.

It is a joint work with Jean Van Schaftingen.

• Chiara Corsato (University of Trieste) — Subharmonic solutions of the prescribed curvature equation.

We study the existence of subharmonic solutions of the capillarity equation $-\Big( u'/{ \sqrt{1+u'^2}}\Big)' = f(t,u).$ According to the behaviour of $$f$$ and to the different nature of the curvature operator at infinity and at zero, we get the existence of arbitrarily large bounded variation subharmonic solutions, or arbitrarily small classical subharmonic solutions. The method of proof relies on the use of non-smooth critical point theory in the former case and on the Poincaré-Birkhoff theorem in the latter one, exploiting some ideas from [3] and [1], respectively. For complete details we refer to [2].

[1] A. Boscaggin and F. Zanolin, Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions, Adv. Nonlinear Stud. 13 (2013), 13–53.
[2] C. Corsato and P. Omari, Subharmonic solutions of the prescribed curvature equation (2013), preprint.
[3] F. Obersnel and P. Omari, The periodic problem for curvature-like equations with asymmetric perturbations, J. Differential Equations, 251 (2011), 1923–1971.
• Fabien Heuwelyckx (UMONS) — Lookback Options in the Binomial Model.

This poster presents how to obtain the price of European lookback options using tree [1]. We study the speed of convergence of such a modelisation to its evaluation with the Black-Scholes model [2] when the length of the tree increases. We show as expected that this convergence is of order $$1/\sqrt{n}$$. To prove this we write the price as an asymptotic expansion in powers of $$1/\sqrt{n}$$ for which we obtain the precise value of the two first coefficients.

[1] Cheuk T.H.F., et Vorst T.C.F. (1997): Currency lookback options and observation frequency: a binomial approach. J. Int. Money Finance 16,173-187.
[2] Goldman M.B., Sosin H.B., et Gatto M.A. (1979): Path dependant options: buy at low, sell at the high. J. Finance 34,1111-1127.
[3] Heuwelyckx F. (submitted): Convergence of european lookback options with floating strike in the binomial model. http://arxiv.org/abs/1302.2312.
• Thomas Kleyntssens (Ulg) — $$S^\nu$$ Spaces, from Theory to Practice.

When considering very irregular functions, it does not make sense to try to characterize the pointwise irregularity because it can change from one point to another. It is more interesting to calculate the spectrum of singularities, ie the size'' of the set of points which share the same pointwise irregularity; by size, one means the Hausdorff dimension.

In practice, computing the spectrum of singularities by using the definition is impossible for real-life signals. One rather uses an indirect way to compute it: the multifractal formalism. The first multifractal formalism was introduced by Frisch and Parisi in the context of fully developped turbulence in [7]. Its main default is that it always leads to a concave spectrum. For this reason, Stéphane Jaffard has introduced in [4] the $$S^\nu$$ spaces. They lead to a new multifractal formalism which can detect non concave spectra.

In this formalism, one needs to calculate the wavelet profile of $$f$$, $\nu(\alpha)=\lim_{\epsilon \to 0^+}\left( \limsup_{j\to +\infty} \left(\frac{\ln(\#E_j(1,\alpha+\epsilon)(f))}{\ln(2^j)} \right) \right)$ where $$E_j(C,\alpha)(f)=\{k : |c_{j,k}| \ge C 2^{-\alpha j}\}$$ and $$(c_{j,k})_{j\in\mathbb{N}, k\in\{0,...,2^j-1\}}$$ are wavelet coefficients of $$f$$.

In practice, one has to avoid the concept of limit and to deal with finite size effects. For example, in theory, the constant 1'' appearing in the definition of $$\nu$$ is arbitrary. But, in practice, one can only calculate a finite number of wavelet coefficients. So, if the constant is too small (resp. too big), the wavelet profile will be equal to the constant function $$1$$ (resp. $$-\infty$$). The choice of this constant becomes capital to determine the correct profile. In this poster, I present a method to determine this constant and I illustrate it numerically on theoretical functions.

[1] J.-M. Aubry, F. Bastin, S. Dispa and S. Jaffard, The spaces $$S^\nu$$ : New spaces defined with wavelet coefficients and related to multifractal analysis, International Journal of Applied Mathematics & Statistics, 2007.
[2] I. Daubechies, Ten lextures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 1992.
[3] S. Jaffard, Multifractal functions: Recent advances and open problems, Bull. Soc. Roy. Sci. Liege 73, 129-153, 2004.
[4] S. Jaffard, Beyond Besov Spaces, part I : Distribution of wavelets coefficients, J. Fourier Anal. Appl. 10, 3, 221–246, 2004.
[5] S. Jaffard and S. Nicolay, Pointwise smoothness of space-filling functions, Applied and Computational Harmonic Analysis 26, 181-199, 2009.
[6] B. Lashermes, Analyse multifractale pratique : coefficients dominants et ordres critiques, Thèse de Doctorat, Ecole Nationale Supérieure de Lyon, 2005.
[7] G. Parisi and U. Frisch, On the Singularity Spectrum of Fully Developed Turbulence, Turbulence and Predictability in Geophysical Fluid Dynamics, Proceedings of the International Summer School in Physics Enrico Fermi, 84-87, 1985.
• Damien Kreit (Ulg) — On Generalized Hölder Spaces.

The Hölder spaces $$C^{\alpha}(\mathbb{R}^{d})$$ ($$\alpha >0$$) provide a natural way for measuring the smoothness of a function. These spaces appear in different areas such as approximation theory and multifractal analysis. The purpose of this poster is to present a generalization of such spaces as well as some recent results about their characterizations ([1,2]). These spaces are a particular case of a generalization of Besov Spaces who have recently been extensively studied ([4]).

[1] D. Kreit, S. Nicolay, Some characterizations of generalized Hölder spaces, Math. Nachr. 285 (2012), 2157-2172.
[2] D. Kreit, S. Nicolay, Characterizations of the elements of generalized Hölder-Zygmund spaces by means of their representation, J. Approx. Theory 172 (2013), 23-36.
[3] D. Kreit, S. Nicolay, Generalized Pointwise Hölder Spaces, submitted.
[4] S. D. Moura, On some characterizations of Besov spaces of generalized smoothness, Math. Nachr. 280 (2007), 1190-1199.
• Ismail Merabet (Université Kasdi Merbah) — Symmetric Interior Penalty Method for Naghdi's Shell.

A symmetric interior penalty method for linearly elastic Naghd's shells in cartesian coordinates is analyzed in this paper. In addition of establishing existence and uniqueness of the discrete problem we derive a posteriori estimator yielding an upper and a lower bounds of the error. A post-processing procedure that can generate $$C^{0}$$ approximate solutions from the discontinuous approximate solution is presented. Numerical tests that validate and illustrate our approach are given.

• Nadine Najdi (UVHC) — On the stability of a weakly thermoelastic Bresse system.

In this work, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. The physical system consists of three wave equations and one heat equation coupled in certain pattern. Only the wave equation about the rotation angle is damped by one locally distributed temperature dissipation law. Then indirect damping is applied to equations of the transverse displacement and the longitudinal displacement of the beam through the coupling terms. Under the equal speed wave propagation condition we show that the system is exponentially stable. Otherwise, in the general case, we establish a new polynomial energy decay rate of type $$1/{\sqrt{t}}$$ under Dirichlet-Neumann-Neumann or Dirichlet-Dirichlet-Dirichlet boundary conditions type.

• Franco Obersnel (Università di Trieste) — Existence and regularity of optimal phase transitions of a one-dimensional model with a saturating flux and a double-well potential.

We will discuss the existence and the regularity of heteroclinic solutions of the prescribed curvature equation $$\label{E1} \Big( u'/{ \sqrt{1+u'^2}}\Big)' = a(t)V'(u).$$ We assume that $$V$$ is a double-well potential, i.e. $$V : \mathbb R \to [0,+\infty[$$ is of class $$C^1$$ and satisfies $$V(x) =0$$ if and only if $$x=\pm 1$$, and $$a\in L^\infty(\mathbb R)$$ is positive. Such an equation finds significant applications in the modeling theory of reaction-diffusion phenomena featuring saturation at large value of the gradient. In particular, an optimal transition between the stable states arises as a minimum of the associated action functional in the space of locally bounded variation functions.

(Joint work with Denis Bonheure and Pierpaolo Omari.)

• David Sbabo (UMONS) — Representations of $$GL_2(\mathbb{Q}_p)$$ arising from elliptic curves over $$\mathbb{Q}_p$$.

In this work, we illustrate the new local p-adic Langlands correspondence for $$GL_2(\mathbb{Q}_p)$$ by P. Colmez ([1]). J-M. Fontaine described the Galois representations of $$\mathbb{Q}_p$$ in terms of etale ($$\varphi,\Gamma$$)-modules and the starting point of the correspondence was a new interpretation of the Fontaine’s rings in terms of p-adic functional analysis. We study the representations of dimension $$2$$ arising from elliptic curves over $$\mathbb{Q}_p$$ and we calculated the associated representations of $$GL_2(\mathbb{Q}_p)$$ in the reducible case.

[1] P. Colmez, Représentations de $$GL_2(\mathbb{Q}_p)$$ et ($$\varphi,\Gamma$$)- modules, Astérisque 330 (2010), pages 281-509
• Zuqi Tang (Université Lille 1) — Residual based a posteriori estimators for the time-harmonic eddy current problems with $$\mathbf{A}/\varphi$$ and $$\mathbf{T}/\Omega$$ potential formulations.

In this poster, the residual based estimators are introduced for the time-harmonic eddy current problems with their classic potential formulations, so-called $$\mathbf{A}/\varphi$$ and $$\mathbf{T}/\Omega$$ potential formulations. The presented estimators evaluate the energy norm error between the exact solution and the numerical one using the finite element method. The reliability and the efficiency of the estimators are presented in [1] and [2]. Some numerical examples are studied in order to show the efficiency of the proposed estimators, which can be used in adaptive mesh refinement.

[1] E. Creusé, S. Nicaise, Z. Tang, Y. Le Menach, N. Nemitz and F. Piriou, Residual-based a posteriori estimators for the $$\mathbf{A}/\varphi$$ magnetodynamic harmonic formulation of the Maxwell system, Mathematical Models and Methods in Applied Sciences, Vol. 22, No. 5, pages. 30, 2012.
[2] E. Creusé, S. Nicaise, Z. Tang, Y. Le Menach, N. Nemitz and F. Piriou, Residual-based a posteriori estimators for the $$\mathbf{T}/\Omega$$ magnetodynamic harmonic formulation of the Maxwell system, International Journal of Numerical Analysis and Modeling, Vol. 10, No. 2, pages. 411-429, 2013.
• Roberta Tittarelli (Université Lille 1) — Space-time residual-based a posteriori estimators for the $$\mathbf A - \varphi$$ formulation of the Maxwell system.

The poster focus on the space-time residual-based a posteriori error estimator for the so-called fully-time $$\mathbf{A} - \varphi$$ potential formulation; this PDE arises from a recast of the Maxwell's equations in the quasi-static approximation by a magnetic vector potential $$\mathbf{A}$$ and an electrical scalar potential $$\varphi$$ on the conductor domain, see e.g. [1]. The proposed error estimator evaluates the numerical error due to the approximation of the above mentioned $$\mathbf{A} - \varphi$$ formulation by the finite element method (in space) and the backward Euler in time. Some numerical results show the space-time adaptivity of the estimator: it can be used to refine the mesh or the time-step in order to get a more accurate solutions (or/and faster computations).

[1] Meunier, Gérard, The finite element method for electromagnetic modeling. ISTE, London, 2008. Translated from the 2008 French original.